The phenomenological theory of sedimentation-consolidation processes of flocculated suspensions is extended to pressure filtration processes. The local mass and linear momentum balances for the solid and liquid component together with appropriate constitutive assumptions lead to a strongly degenerate (mixed hyperbolic-parabolic) nonlinear partial differential equation for the local solids fraction, which together with initial and boundary conditions determines a dynamic cake filtration process. In the case of a prescribed applied pressure function, we obtain a free boundary problem, in which the piston height has to be determined simultaneously with the solids concentration, A numerical algorithm approximating the physically correct solution, with possible discontinuities such as the cake/suspension interface, is presented and employed to simulate various cake filtration processes.